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Properties

Label 1-373-373.124-r0-0-0
Degree $1$
Conductor $373$
Sign $0.287 + 0.957i$
Analytic cond. $1.73220$
Root an. cond. $1.73220$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.638 − 0.769i)2-s + (−0.117 − 0.993i)3-s + (−0.184 − 0.982i)4-s + (−0.713 + 0.701i)5-s + (−0.839 − 0.543i)6-s + (−0.440 + 0.897i)7-s + (−0.874 − 0.485i)8-s + (−0.972 + 0.234i)9-s + (0.0843 + 0.996i)10-s + (0.0168 − 0.999i)11-s + (−0.954 + 0.299i)12-s + (−0.874 + 0.485i)13-s + (0.409 + 0.912i)14-s + (0.780 + 0.625i)15-s + (−0.931 + 0.363i)16-s + (−0.758 + 0.651i)17-s + ⋯
L(s)  = 1  + (0.638 − 0.769i)2-s + (−0.117 − 0.993i)3-s + (−0.184 − 0.982i)4-s + (−0.713 + 0.701i)5-s + (−0.839 − 0.543i)6-s + (−0.440 + 0.897i)7-s + (−0.874 − 0.485i)8-s + (−0.972 + 0.234i)9-s + (0.0843 + 0.996i)10-s + (0.0168 − 0.999i)11-s + (−0.954 + 0.299i)12-s + (−0.874 + 0.485i)13-s + (0.409 + 0.912i)14-s + (0.780 + 0.625i)15-s + (−0.931 + 0.363i)16-s + (−0.758 + 0.651i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.287 + 0.957i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.287 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1401796540 + 0.1042736172i\)
\(L(\frac12)\) \(\approx\) \(0.1401796540 + 0.1042736172i\)
\(L(1)\) \(\approx\) \(0.6747782222 - 0.3980796365i\)
\(L(1)\) \(\approx\) \(0.6747782222 - 0.3980796365i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.638 - 0.769i)T \)
3 \( 1 + (-0.117 - 0.993i)T \)
5 \( 1 + (-0.713 + 0.701i)T \)
7 \( 1 + (-0.440 + 0.897i)T \)
11 \( 1 + (0.0168 - 0.999i)T \)
13 \( 1 + (-0.874 + 0.485i)T \)
17 \( 1 + (-0.758 + 0.651i)T \)
19 \( 1 + (0.151 + 0.988i)T \)
23 \( 1 + (0.918 - 0.394i)T \)
29 \( 1 + (-0.315 + 0.948i)T \)
31 \( 1 + (-0.954 - 0.299i)T \)
37 \( 1 + (-0.999 + 0.0337i)T \)
41 \( 1 + (-0.0506 + 0.998i)T \)
43 \( 1 + (-0.184 - 0.982i)T \)
47 \( 1 + (0.585 + 0.810i)T \)
53 \( 1 + (-0.972 - 0.234i)T \)
59 \( 1 + (-0.664 - 0.747i)T \)
61 \( 1 + (-0.117 - 0.993i)T \)
67 \( 1 + (-0.250 - 0.968i)T \)
71 \( 1 + (0.943 - 0.331i)T \)
73 \( 1 + (0.283 + 0.959i)T \)
79 \( 1 + (0.0843 + 0.996i)T \)
83 \( 1 + (-0.972 - 0.234i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.758 + 0.651i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.33225005176939721243143260190, −23.38289941728313047939920237388, −22.76124127412013056459893816545, −22.152157268092503980864058046699, −20.919022572794326632176351990612, −20.26504188157872014239891242096, −19.6100205560876623827990294755, −17.62021963539004873035949594548, −17.141131229527354675926563365850, −16.28978586686274059625958383981, −15.46002304781878940031420232154, −15.01853646415507327680161150247, −13.753121628726338475072029132464, −12.86224704117881527195144955149, −11.934390703831284577291969936161, −10.93591125791897081521964851595, −9.59766225422623555044439392114, −8.9022513205605302596426014087, −7.54705074313079693350873335757, −6.95103089081962760705828389453, −5.32171296279916183918104749423, −4.672960352292913334361461961517, −3.9566897077599506040617694895, −2.853826146644244933396319999664, −0.08111354101986912490540840478, 1.77092279583785567356738432159, 2.79686534452842414394958122835, 3.59369682549731281067057966682, 5.1653071786388440121874073676, 6.18990171451858063295863436052, 6.907018657740651416106503602587, 8.2826301397235110604448508889, 9.26718377564897090336144616462, 10.76235029012236892856375843087, 11.36121131965523707260770477562, 12.31728981306568528985396952215, 12.78372879325201060726326533417, 14.0501682975491302619410720767, 14.66044494574127419014501713739, 15.64303663948489131650657968886, 16.88397375254195693671181909462, 18.37278460982701278206690832518, 18.79682351908609422411454352110, 19.36954936543037785116304290426, 20.172774533282927102316413823382, 21.61182726589415421563292474354, 22.2244283224738213207637653669, 22.900388460389981691252686635027, 23.94473499160625073294166472045, 24.36778397548679496555343805526

Graph of the $Z$-function along the critical line